Equality-constrained minimization of polynomial functions

Abstract

This paper investigates the equality-constrained minimization of polynomial functions. Let ℝ be the field of real numbers, and ℝ[x1, …, xn] the ring of polynomials over ℝ in variables x1, …, xn. For an f ∈ ℝ[x1, …, xn] and a finite subset H of ℝ[x1, …, xn], denote by \(\mathcal{V}(f:H)\) the set {\(\left. {f\left( {\bar \alpha } \right)} \right|\bar \alpha \in \mathbb{R}^n\), and \(h(\bar \alpha ) = 0,\forall h \in H\)}. We provide an effective algorithm for computing a finite set U of non-zero univariate polynomials such that the infimum \(inf \mathcal{V}(f:H)\) of \(\mathcal{V}(f:H)\) is a root of some polynomial in U whenever \(inf \mathcal{V}(f:H) \ne \pm \infty\). The strategies of this paper are decomposing a finite set of polynomials into triangular chains of polynomials and computing the so-called revised resultants. With the aid of the computer algebraic system Maple, our algorithm has been made into a general program to treat the equality-constrained minimization of polynomials with rational coefficients.